Question

The velocity function and initial position of Runners \(A\) and B are given. Analyze the race that results by graphing the position functions of the runners and finding the time and positions (if any) at which they first pass each other. $$\mathbf{A}: v(t)=\sin t ; s(0)=0 \quad \mathbf{B}: V(t)=\cos t ; S(0)=0$$

Short answer

Answer: The runners first pass each other when $$t = \frac{\pi}{2}$$ and at a position of 1.

Step-by-step solution

Integrate the velocity functions

First, we need to find the position functions for both runners A and B. To do this, we will integrate their respective velocity functions: For runner A: $$s_A(t) = \int v(t) dt = \int \sin t dt = -\cos(t) + C_A$$ For runner B: $$s_B(t) = \int V(t) dt = \int \cos t dt = \sin(t) + C_B$$ Now, we need to find the constants of integration (\(C_A\) and \(C_B\)) using the initial positions: For runner A: $$s_A(0) = -\cos(0) + C_A = 0 \Rightarrow C_A = 1$$ For runner B: $$s_B(0) = \sin(0) + C_B = 0 \Rightarrow C_B = 0$$ So the position functions are: $$s_A(t) = -\cos(t) + 1$$ $$s_B(t) = \sin(t)$$

Find when they pass each other

To find when the runners pass each other, we need to set their position functions equal to each other and solve for t: $$s_A(t) = s_B(t)$$ $$-\cos(t)+1 = \sin(t)$$ $$\sin(t) + \cos(t) = 1$$ Using the substitution \(u = \sin(t)\): $$u + \sqrt{1 - u^2} = 1$$ $$\sqrt{1 - u^2} = 1 - u$$ $$1 - u^2 = 1 - 2u + u^2$$ $$2u^2 - 2u = 0$$ $$ 2u(u - 1) = 0$$ From this, the solutions are \(u = 0\) or \(u = 1\). Substituting back, we get the time values: $$t_1 = \arcsin(0) = 0$$ $$t_2 = \arcsin(1) = \frac{\pi}{2}$$ Since we are interested in when they first pass, we will use the nonzero time, $$t = \frac{\pi}{2}$$.

Find the positions during passing

Using \(t = \frac{\pi}{2}\), we will find the position of both runners A and B. We can use either runner's position function, as they should be equal: $$s_A\left(\frac{\pi}{2}\right) = -\cos\left(\frac{\pi}{2}\right) + 1 = 1$$ So, both runners first pass each other at a position of 1 when $$t = \frac{\pi}{2}$$.

Graph the position functions

Finally, plot the position functions for both runners A and B on the same graph to analyze the race: $$s_A(t) = -\cos(t) + 1 \text{ and } s_B(t) = \sin(t)$$ From the graph, we can see how the runners' positions change over time. The two curves intersect at the point $$(\frac{\pi}{2}, 1)$$, representing the moment and position when they first pass each other. This confirms our previous result.

Key concepts

Velocity Functions

Velocity functions describe how fast something is moving and in what direction. They are crucial for understanding motion. In the example provided, the velocity functions for runners A and B are given by \(v(t) = \sin t\) and \(V(t) = \cos t\), respectively. These functions tell us how each runner’s speed varies over time. A positive velocity indicates forward motion, whereas a negative velocity indicates backward motion.

To transition from a velocity function to a position function, one must integrate the velocity function. This process provides the distance covered over time, incorporating any directional changes. It’s like adding up all the little bits of distance covered to find the total distance at any given time.

Understanding velocity functions is essential because they serve as the foundation for deriving position functions, which are needed to analyze the motion of objects or people over time.

Graphing Functions

Graphing functions is a way to visually interpret mathematical equations. It lets us see how values change over time and helps to identify key points such as where runners might pass each other. In this case, the position functions \(s_A(t) = -\cos(t) + 1\) and \(s_B(t) = \sin(t)\) are graphed to analyze the race between runners A and B.

When graphing a function, important aspects to consider include:

• Intercepts: Points where the graph crosses the axes.
• Maxima and Minima: The peaks and valleys in the graph.
• Intersection Points: Key for understanding when two functions are equal, like when runners A and B pass each other.

Plotting these position functions shows that the runners pass at \(t = \frac{\pi}{2}\), a critical moment for analyzing their relative positions. Graphing makes mathematical concepts intuitive by transforming abstract formulas into visual pictures.

Integration

Integration is a fundamental concept in calculus used to find position functions from velocity functions. It is the process of finding a function when its derivative is known, essentially undoing differentiation. In the original exercise, integrating the velocity functions of the runners transforms them into position functions.

For instance, given the velocity function for runner A, \(v(t) = \sin t\), integration yields the position function \(s_A(t) = \int \sin t \, dt = -\cos(t) + C_A\). Likewise, for runner B, \(V(t) = \cos t\) becomes \(s_B(t) = \int \cos t \, dt = \sin(t) + C_B\).

Determining the constants of integration, \(C_A\) and \(C_B\), is done using given conditions such as initial positions \(s(0) = 0\). This step ensures the integration accurately reflects the scenario. By mastering integration, you can solve many real-world problems involving changes over time.